Ex 1.1, 8 (Introduction)
Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a,b):|a b| is even} , is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
Modulus function
1 = 1
2 = 2
0 = 0
1 = 1
3 = 3
Ex 1.1, 8
Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a,b):|a b| is even} , is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
R = {(a, b):|a b| is even} where a, b A
Check Reflexive
Since |a a| = |0| = 0
& 0 is always even
|a a| is even
(a, a) R,
R is reflexive.
Check symmetric
We know that
|a b| = |b a|
Hence, if |a b| is even,
then |b a| is also even
Hence, If (a, b) R, then (b, a) R
R is symmetric
Check transitive
If |a b| is even , then (a b) is even
Similarly, if |b c| is even , then (b c) is even
Now,
Sum of even numbers is also even
a b + b c is even
a c is even
Hence, |a c| is even
So, If |a b| &|b c| is even , then |a c| is even
i.e. If (a, b) R & (b, c) R , then (a, c) R
R is transitive
Since R is reflexive, symmetric and transitive,
it is equivalence relation
R = {(a,b):|a b| is even}
Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
In {1, 3, 5},
All elements are odd,
So, difference between any 2 numbers is always even
Hence,
Modulus of difference between any 2 numbers is always even
Hence, element of {1, 3, 5} are related to each other
In {2, 4},
All elements are even,
So, difference between any 2 numbers is always even
Hence,
Modulus of difference between any 2 numbers is always even
Hence, element of {2, 4} are related to each other
In {1, 3, 5} & {2, 4},
Elements of {1, 3, 5} are odd
Elements of {2, 4} are even
Difference of one element from {1, 3, 5} and one element from {2, 4} is odd
As Difference of even and odd number is always odd
Difference is not even
If difference not even,
Modulus of difference also not even
Hence, element of {1, 3, 5} & {2, 4} are not related to each other

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.